Integrable Lattices: Random Matrices and Random Permutations

Abstract

These lectures present a survey of recent developments in the area of random matrices (finite and infinite) and random permutations. These probabilistic problems suggest matrix integrals (or Fredholm determinants), which arise very naturally as integrals over the tangent space to symmetric spaces, as integrals over groups and finally as integrals over symmetric spaces. An important part of these lectures is devoted to showing that these matrix integrals, upon apropriately adding time-parameters, are natural tau-functions for integrable lattices, like the Toda, Pfaff and Toeplitz lattices, but also for integrable PDE's, like the KdV equation. These matrix integrals or Fredholm determinants also satisfy Virasoro constraints, which combined with the integrable equations lead to (partial) differential equations for the original probabilities.

References in corpus (3)

• Fredholm Determinants, Differential Equations and Matrix Models
• The spectrum of random matrices and integrable systems
• The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff

Cited by in corpus (9)

• Distribution functions for largest eigenvalues and their applications
• Random Words, Toeplitz Determinants and Integrable Systems. II
• Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions
• Nonperturbative Superpotentials and Compactification to Three Dimensions
• Tau Functions and Matrix Integrals
• Stampedes II: Null Polygons in Conformal Gauge Theory
• Symmetric Matrix Ensemble and Integrable Hydrodynamic Chains
• On $\tau$-function of conjugate nets
• Painlevé expressions for LOE, LSE and interpolating ensembles